In this video I go over further into the Arc Length Formula and this time explain how it is sometimes useful to have a function that measures the arc length of a curve from a particular starting point point to any other point on the curve. From the arc length formula we determine this function, which is called the Arc Length Function. Also in this video, I write the integrand of the arc length formula in terms of differentials in order to develop a useful visual geometric interpretation of the rate of change of arc length. This is a very interesting video on arc length so make sure to watch this video!
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Applications of Integrals: Arc Length Function
It is often times useful to have a function that measures the arc length of a curve from a particular starting point to any other point on the curve.
Thus, if a smooth curve C has the equation y = f(x) for a ≤ x ≤ b, let s(x) be the distance along C from the initial point P0(a, f(a)) to the point Q(x, f(x)). Then s is a function called the Arc Length Function and, from the Arc Length Formula:
Note: We have replaced the variable of integration x so that x does not have two meanings.
We can use Part 1 of the Fundamental Theorem of Calculus to differentiate s(x), since the integrand is continuous:
This shows that the rate of change of s with respect to x is always at least 1 and is equal to 1 when f'(x), the slope of the curve, is 0.
The differential of arc length is:
And this is sometimes written in the symmetric form:
The geometric interpretation of the above equation is shown below:
This interpretation can be used as a mnemonic device, i.e. memory aid, to remember the arc length formula when using the Pythagorean theorem:
We can also work backwards to solve for the Arc Length Formula when dealing with x as a function of y instead:
